Using residue calculus, Siegel presented the transformation law of the Dedekind eta function under the inversion action \( \cite {Siegel^{\prime }s method}\) . Inspired by Siegel, Raji proved in \( \cite {Raji}\text{ }\) the transformation law under the inversion of \(\theta _3(z,\tau )\) . In this paper, we determine two functions on which we apply residue calculus to prove the transformation law of \( \theta _1 \)  and \( \theta _3 \)  functions under the generators in (6) and (7) respectively. The transformation law of the generator of \(\Gamma (2)\)  has not been explicitly proven for \(\theta _3\)  where our proof is different from that of Raji's in a sense that we determine the transformation law of \(\theta _3\)  under the action of the generator of \(\Gamma (2)\)  not the inversion. In this case \(\theta _3\)  transforms into itself. The remaining theta function \(\theta _2\)  and \(\theta _4\)  are related to \(\theta _1\)  and \(\theta _3\)  via relation described in [1]}, hence we will not address the proof of these two functions.
